Helmut Farbmacher

The many weak instruments problem and Mendelian randomization

Instrumental variable estimates of causal effects can be biased when using many instruments that are only weakly associated with the exposure. We describe several techniques to reduce this bias and estimate corrected standard errors. We present our findings using a simulation study and an empirical application. For the latter, we estimate the effect of height on lung function, using genetic variants as instruments for height. Our simulation study demonstrates that, using many weak individual variants, two-stage least squares (2SLS) is biased, whereas the limited information maximum likelihood (LIML) and the continuously updating estimator (CUE) are unbiased and have accurate rejection frequencies when standard errors are corrected for the presence of many weak instruments. Our illustrative empirical example uses data on 3631 children from England. We used 180 genetic variants as instruments and compared conventional ordinary least squares estimates with results for the 2SLS, LIML, and CUE instrumental variable estimators using the individual height variants. We further compare these with instrumental variable estimates using an unweighted or weighted allele score as single instruments. In conclusion, the allele scores and CUE gave consistent estimates of the causal effect. In our empirical example, estimates using the allele score were more efficient. CUE with corrected standard errors, however, provides a useful additional statistical tool in applications with many weak instruments. The CUE may be preferred over an allele score if the population weights for the allele score are unknown or when the causal effects of multiple risk factors are estimated jointly.

Heterogeneous Effects of a Nonlinear Price Schedule for Outpatient Care

Nonlinear price schedules generally have heterogeneous effects on health-care demand. We develop and apply a finite mixture bivariate probit model to analyze whether there are heterogeneous reactions to the introduction of a nonlinear price schedule in the German statutory health insurance system. In administrative insurance claims data from the largest German health insurance plan, we find that some individuals strongly react to the new price schedule while a second group of individuals does not react. Post-estimation analyses reveal that the group of the individuals who do not react to the reform includes the relatively sick. These results are in line with forward-looking behavior: Individuals who are already sick expect that they will hit the kink in the price schedule and thus are less sensitive to the co-payment. Copyright

On the Use of the Lasso for Instrumental Variables Estimation with Some Invalid Instruments

ABSTRACT We investigate the behavior of the Lasso for selecting invalid instruments in linear instrumental variables models for estimating causal effects of exposures on outcomes, as proposed recently by Kang et al. Invalid instruments are such that they fail the exclusion restriction and enter the model as explanatory variables. We show that for this setup, the Lasso may not consistently select the invalid instruments if these are relatively strong. We propose a median estimator that is consistent when less than 50% of the instruments are invalid, and its consistency does not depend on the relative strength of the instruments, or their correlation structure. We show that this estimator can be used for adaptive Lasso estimation, with the resulting estimator having oracle properties. The methods are applied to a Mendelian randomization study to estimate the causal effect of body mass index (BMI) on diastolic blood pressure, using data on individuals from the UK Biobank, with 96 single nucleotide polymorphisms as potential instruments for BMI. Supplementary materials for this article are available online.

EXTENSIONS OF HURDLE MODELS FOR OVERDISPERSED COUNT DATA

Hurdle models are frequently used to model count data. Recent developments in the count data literature make it possible to relax commonly imposed assumptions of these models. On the basis of these findings, two extensions of hurdle models that make popular specifications more flexible are developed. Both extensions nest the models that have been used so far, so they can be tested by appropriate parametric restrictions. An example from health economics illustrates the relevance of both model extensions.

Inference Problems Under a Special Form of Heteroskedasticity

In the presence of heteroskedasticity, conventional standard errors (which assume homoskedasticity) can be biased up or down. The most common form of heteroskedasticity leads to conventional standard errors that are too small. When Wald tests based on these standard errors are insignificant, heteroskedasticity ro- bust standard errors do not change inference. On the other hand, inference is conservative in a setting with upward-biased conventional standard errors. We discuss the power gains when using robust standard errors in this case and also potential problems of heteroskedasticity tests. As a solution for the poor performance of the usual heteroskedasticity tests in this setting, we propose a modification of the White test which has better properties. We illustrate our findings using a study in labor economics. The correct standard errors turn out to be around 15 percent lower, leading to different policy conclusions. Moreover, only our modified test is able to detect heteroskedasticity in this application.

Testing under a special form of heteroscedasticity

ABSTRACT In the presence of heteroscedasticity, conventional standard errors (which assume homoscedasticity) can be biased up or down. The most common form of heteroscedasticity leads to conventional standard errors that are too small. In this study, we discuss the conditions under which conventional standard errors are too large. In such settings, standard tests of heteroscedasticity may fail and leave the heteroscedasticity undetected. This is particularly problematic as power gains can be achieved when testing for the causal effect in such settings.